Density functional theory Koopmans theorem

Gritsenko O, Baerends EJ (2002) The analog of Koopmans theorem in spin-density functional theory, J Chem Phys, 117 9154-9159... 

A popular alternative is to employ density functional theory (DFT) methods. Kohn-Sham (KS) orbital energies in the ground state are a more reliable predictor of the ion state ordering than Koopmans theorem. There are good theoretical reasons for interpreting them as approximate vertical IE." For transition metal compounds AE methods, the DFT equivalent... 

Koopmans Theorem applies to Hartree-Fock theory by virtue of the particular method for evaluating the quantum mechanical exchange interaction. In Density Functional Theory, a different method is employed. Hence, HF orbitals are not the same as DFT orbitals and Koopmans Theorem does not apply. This can be illustrated with reference to Slater s Xu (i.e. DFT exchange only) model [15]. 

The most promising approaches for efficient electronic structure calculations on large molecules are generally based on density functional theory with Kohn-Sham orbitals [32-35]. The most efficient such method for CE-BEs is based on Koopmans theorem, but this approach has quite limited accuracy [36-39]. Better accuracy can be obtained from calculations based on an effective core potential [40-45], an equivalent core approximation [46-48], a fractionally occupied transition state [49-52], or with a ASCF approach [29, 31, 53-57]. Time-dependent density functional theory is also widely used for CEBE calculation [58-62], wherein the best results are usually given with functionals having a long-range correction [63, 64]. 

Basing on the first principles of Quantum mechanics as exposed in the previous chapters and sections, special chapters of quantum theory are here unfolded in order to further extend and caching the quantum information from free to observed evolution within the matter systems with constraints (boundaries). As such, the Feynman path integral formalism is firstly exposed and then applied to atomic, quantum barrier and quantum harmonically vibration, followed by density matrix approach, opening the Hartree-Fock and Density Functional pictures of many-electronic systems, with a worthy perspective of electronic occupancies via Koopmans theorem, while ending with a further generalization of the Heisenberg observability and of its first application to mesosystems. 

The next five chapters are each devoted to the study of one particular computational model of ab initio electronic-structure theory Chapter 10 is devoted to the Hartree-Fock model. Important topics discussed are the parametrization of the wave function, stationary conditions, the calculation of the electronic gradient, first- and second-order methods of optimization, the self-consistent field method, direct (integral-driven) techniques, canonical orbitals, Koopmans theorem, and size-extensivity. Also discussed is the direct optimization of the one-electron density, in which the construction of molecular orbitals is avoided, as required for calculations whose cost scales linearly with the size of the system. 

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